In this appendix, we derive some properties of the classical one-liquid dip-coating flow [1]} that are relevant for the discussion of its two-liquid counterpart.
For a single-phase LLD flow, the film thickness \(\hat{h}\)  obeys the differential equation
\(\hat{h}_\infty = \hat{h}+ s \hat{h}^{\prime \prime \prime }\frac{\hat{h}^3}{3},\) 
